3.2.19 \(\int \frac {(a+b x)^7}{x^{13}} \, dx\)

Optimal. Leaf size=96 \[ -\frac {b^4 (a+b x)^8}{3960 a^5 x^8}+\frac {b^3 (a+b x)^8}{495 a^4 x^9}-\frac {b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}-\frac {(a+b x)^8}{12 a x^{12}} \]

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Rubi [A]  time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {45, 37} \begin {gather*} -\frac {b^4 (a+b x)^8}{3960 a^5 x^8}+\frac {b^3 (a+b x)^8}{495 a^4 x^9}-\frac {b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}-\frac {(a+b x)^8}{12 a x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^13,x]

[Out]

-(a + b*x)^8/(12*a*x^12) + (b*(a + b*x)^8)/(33*a^2*x^11) - (b^2*(a + b*x)^8)/(110*a^3*x^10) + (b^3*(a + b*x)^8
)/(495*a^4*x^9) - (b^4*(a + b*x)^8)/(3960*a^5*x^8)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {(a+b x)^7}{x^{13}} \, dx &=-\frac {(a+b x)^8}{12 a x^{12}}-\frac {b \int \frac {(a+b x)^7}{x^{12}} \, dx}{3 a}\\ &=-\frac {(a+b x)^8}{12 a x^{12}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}+\frac {b^2 \int \frac {(a+b x)^7}{x^{11}} \, dx}{11 a^2}\\ &=-\frac {(a+b x)^8}{12 a x^{12}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}-\frac {b^2 (a+b x)^8}{110 a^3 x^{10}}-\frac {b^3 \int \frac {(a+b x)^7}{x^{10}} \, dx}{55 a^3}\\ &=-\frac {(a+b x)^8}{12 a x^{12}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}-\frac {b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac {b^3 (a+b x)^8}{495 a^4 x^9}+\frac {b^4 \int \frac {(a+b x)^7}{x^9} \, dx}{495 a^4}\\ &=-\frac {(a+b x)^8}{12 a x^{12}}+\frac {b (a+b x)^8}{33 a^2 x^{11}}-\frac {b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac {b^3 (a+b x)^8}{495 a^4 x^9}-\frac {b^4 (a+b x)^8}{3960 a^5 x^8}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 93, normalized size = 0.97 \begin {gather*} -\frac {a^7}{12 x^{12}}-\frac {7 a^6 b}{11 x^{11}}-\frac {21 a^5 b^2}{10 x^{10}}-\frac {35 a^4 b^3}{9 x^9}-\frac {35 a^3 b^4}{8 x^8}-\frac {3 a^2 b^5}{x^7}-\frac {7 a b^6}{6 x^6}-\frac {b^7}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^13,x]

[Out]

-1/12*a^7/x^12 - (7*a^6*b)/(11*x^11) - (21*a^5*b^2)/(10*x^10) - (35*a^4*b^3)/(9*x^9) - (35*a^3*b^4)/(8*x^8) -
(3*a^2*b^5)/x^7 - (7*a*b^6)/(6*x^6) - b^7/(5*x^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^7}{x^{13}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^7/x^13,x]

[Out]

IntegrateAlgebraic[(a + b*x)^7/x^13, x]

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fricas [A]  time = 1.13, size = 79, normalized size = 0.82 \begin {gather*} -\frac {792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^13,x, algorithm="fricas")

[Out]

-1/3960*(792*b^7*x^7 + 4620*a*b^6*x^6 + 11880*a^2*b^5*x^5 + 17325*a^3*b^4*x^4 + 15400*a^4*b^3*x^3 + 8316*a^5*b
^2*x^2 + 2520*a^6*b*x + 330*a^7)/x^12

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giac [A]  time = 1.15, size = 79, normalized size = 0.82 \begin {gather*} -\frac {792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^13,x, algorithm="giac")

[Out]

-1/3960*(792*b^7*x^7 + 4620*a*b^6*x^6 + 11880*a^2*b^5*x^5 + 17325*a^3*b^4*x^4 + 15400*a^4*b^3*x^3 + 8316*a^5*b
^2*x^2 + 2520*a^6*b*x + 330*a^7)/x^12

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maple [A]  time = 0.01, size = 80, normalized size = 0.83 \begin {gather*} -\frac {b^{7}}{5 x^{5}}-\frac {7 a \,b^{6}}{6 x^{6}}-\frac {3 a^{2} b^{5}}{x^{7}}-\frac {35 a^{3} b^{4}}{8 x^{8}}-\frac {35 a^{4} b^{3}}{9 x^{9}}-\frac {21 a^{5} b^{2}}{10 x^{10}}-\frac {7 a^{6} b}{11 x^{11}}-\frac {a^{7}}{12 x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^13,x)

[Out]

-1/5*b^7/x^5-7/6*a*b^6/x^6-35/8*a^3*b^4/x^8-7/11*a^6*b/x^11-3*a^2*b^5/x^7-35/9*a^4*b^3/x^9-21/10*a^5*b^2/x^10-
1/12*a^7/x^12

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maxima [A]  time = 1.45, size = 79, normalized size = 0.82 \begin {gather*} -\frac {792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^13,x, algorithm="maxima")

[Out]

-1/3960*(792*b^7*x^7 + 4620*a*b^6*x^6 + 11880*a^2*b^5*x^5 + 17325*a^3*b^4*x^4 + 15400*a^4*b^3*x^3 + 8316*a^5*b
^2*x^2 + 2520*a^6*b*x + 330*a^7)/x^12

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mupad [B]  time = 0.07, size = 79, normalized size = 0.82 \begin {gather*} -\frac {\frac {a^7}{12}+\frac {7\,a^6\,b\,x}{11}+\frac {21\,a^5\,b^2\,x^2}{10}+\frac {35\,a^4\,b^3\,x^3}{9}+\frac {35\,a^3\,b^4\,x^4}{8}+3\,a^2\,b^5\,x^5+\frac {7\,a\,b^6\,x^6}{6}+\frac {b^7\,x^7}{5}}{x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/x^13,x)

[Out]

-(a^7/12 + (b^7*x^7)/5 + (7*a*b^6*x^6)/6 + (21*a^5*b^2*x^2)/10 + (35*a^4*b^3*x^3)/9 + (35*a^3*b^4*x^4)/8 + 3*a
^2*b^5*x^5 + (7*a^6*b*x)/11)/x^12

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sympy [A]  time = 0.80, size = 85, normalized size = 0.89 \begin {gather*} \frac {- 330 a^{7} - 2520 a^{6} b x - 8316 a^{5} b^{2} x^{2} - 15400 a^{4} b^{3} x^{3} - 17325 a^{3} b^{4} x^{4} - 11880 a^{2} b^{5} x^{5} - 4620 a b^{6} x^{6} - 792 b^{7} x^{7}}{3960 x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**13,x)

[Out]

(-330*a**7 - 2520*a**6*b*x - 8316*a**5*b**2*x**2 - 15400*a**4*b**3*x**3 - 17325*a**3*b**4*x**4 - 11880*a**2*b*
*5*x**5 - 4620*a*b**6*x**6 - 792*b**7*x**7)/(3960*x**12)

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